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work in invariants was held by his contemporaries may, however,
be indicated by Sylvester's characteristic remark that 'Cayley,
Hennite, and I constitute an Invariantive Trinity.' Who was
who in this astounding trinity Sylvester omitted to state; but
perhaps this oversight is immaterial, as each member of such a
trefoil would be capable of transforming himself into himself or
into either of his coinvariantive beings.
The two fields in which Hennite found what are perhaps the
most striking individual results in all his beautiful work are
those of the general equation of the fifth degree and transcen-
dental numbers. The nature of what he found in the first is
clearly indicated in the introduction to his short note Sur la
resolution de Vequation du cinquieme degre (On the Solution of
the [general] Equation of the Fifth Degree; published in the
Comptes rendus de IAcademic des Sciences for 18589 when
Hermite was thirty-six).
'It is known that the general equation of the fifth degree can
be reduced, by a substitution [on the unknown x] whose coeffi-
cients are determined without using any irrationalities other
than square roots or cube roots, to the form
x5  x  a = 0.
[That is, if we can solve this equation for x, then we can solve the
general equation of the fifth degree.]
"This remarkable result, due to the English mathematician
Jerrard, is the most important step that has been taken in the
algebraic theory of equations of the fifth degree since Abel
proved that a solution by radicals is impossible. This impossi-
bility shows in fact the necessity for introducing some new
analytic element [some new kind of function] in seeking the
solution, and, on this account, it seems natural to take as an
auxiliary the roots of the very simple equation we have just
mentioned. Nevertheless, in order to legitimize its use rigor-
ously as an essential element in the solution of the general equa-
tion, it remains to see if this simplicity of form actually permits
us to arrive at some idea of the nature of its roots, to grasp
what is peculiar and essential in the mode of existence of these
quantities, of which nothing is known beyond the fact that they
are not expressible by radicals.